where the remainder term \(R(P,Q)\)  is controlled.
For example, if we apply equation (REF ) to \(Q=\widehat{P}_n\)  the empirical distribution, the influence function provides a first order approximation for the difference between the estimator \(T(\widehat{P}_n)=T(X_1^n)\)  and its limit \(T(P)\) .
This technique of approximating the estimator by its influence function is also linked to the Bahadur decomposition, see [1]} and [2]} for applications to M-estimators.
The influence function of \(M\) -estimators is usually chosen bounded in robust statistics, in particular from [3]}, [4]} we have that if \(\psi \)  is bounded, then the influence function is bounded and \(T\)  is qualitatively robust (i.e. the estimator \(T(X_1^n)\)  is equi-continuous, c.f. [4]}) and have asymptotic breakdown point \(1/2\) . On the other hand if \(\psi \)  is unbounded, then \(T(X_1^n)\)  is not qualitatively robust, the influence function is not bounded and the asymptotic breakdown point is zero. From Hampel's Theorem [4]} we also have that \(\psi \)  is bounded if and only if \(T\)  is a continuous functional with respect to the Levy metric. More generally, the influence function has been used in a lot of works on asymptotic robustness, see [7]}, [4]} or [9]}, [10]}.
